The present invention relates to robust information extraction, and more particularly to a filtering process for stable and accurate estimation.
In most information extraction applications, propagation of an internal state of a dynamic system, which is not measurable by sensors directly, is naturally described by nonlinear continuous-time differential equations, based upon physical laws,
x(t)=f(x(t))+w(t)
where x(t) is a state vector, f(x) is a vector nonlinear function describing the system dynamics, and w(t) is a system noise with a covariance matrix Q(t).
Measurable quantities, which are generally available at intermittent discrete times, are usually in nonlinear relationship with the internal system states,
z(k)=h(x(k))+v(k)
where z(k) represents a discrete measurement at the k-th sampling time tk, h(x) is a vector nonlinear function which describes the relationship of the measurement with the state vector, and v(k) is a measurement noise with a covariance of R(k).
A filtering process deals with the problem of extracting the internal, sometimes hidden and unmeasurable, state, x(t), from the measurement z(k). Applications of the filtering process are very common, to name a few, such as classifying the component materials from hyperspectral imagery, obtaining the aircraft position and attitude from the accelerometer and gyro measurements of an Inertial Measurement Unit, and tracking a target""s velocity and acceleration from a Radar""s positional measurements.
Under the conditions of a linear system with linear measurements, a Gaussian system noise and a Gaussian measurement noise, a linear Kalman filter provides an optimal estimate of the internal system state. The estimate is optimal in the sense that the covariance of the estimate is minimal, the residuals are a white Gaussian noise process, and innovative information is absent in the residuals. If selectively, the system model and the measurement model are nonlinear, extracting an estimate of the internal system state x(t), from noisy measurements, z(k), is conventionally accomplished by an extended Kalman filter, and selectively, in some cases, by a variant of the extended Kalman filter. An extended Kalman filter shares most of the salient features with a linear Kalman filter, by linearizing the nonlinear system equations and measurement equations about the most recent estimate and taking a first-order approximation to a Taylor-series expansion.
In the nonlinear cases, an optimal estimate of the system state is difficult for the extended Kalman filter to attain. A direct implementation of an extended Kalman filter possesses several inherent drawbacks. First, it is prone to numerical divergence. The correct propagation of the state estimates relies on the proper propagation of a covariance error matrix, which must remain symmetric and positive definite all the time, due to its mathematical definition. Finite-length manipulations, such as computer roundoff error, usually result in a loss of the symmetry and positive definiteness of the covariance error matrix resulting in numerical instability. A feasible approach is unavailable to recover from such a numerical divergence status.
An extended Kalman filter passively accepts measurements and is sensitive to measurement quality. Conventionally, it accepts whatever measurement data the measurement input interface provides in full confidence. The Kalman filter is unable to distinguish good-quality measurements from low-quality ones. It generally does not attempt to correct the measurement data by any means even though there might be enough evidence that the quality of measurement data is very poor. When the quality of the measurements applied to the Kalman filter is worse than that accounted for by the measurement noise, the results generated by the Kalman filter are usually meaningless, and yet, it is difficult to determine whether the results are meaningless.
Another difficulty in a conventional extended Kalman filter is the requirement of discretizing the continuous-time differential model into a discrete-time difference model. This in turn requires the selection of a discretization time prior to the design of the extended Kalman filter. A standard method is to set the discretization time the same as the sampling period of the measurements. Such a prediscretization approach does not guarantee that the discrete-time difference model is a good approximation to the continuous-time differential model, especially in the case of a slow sampling rate. Neither is this prediscretization approach able to detect a divergence of the discrete-time difference model from the continuous-time differential model when the divergence occurs.
An objective of the present invention is to provide a filtering process for the most general formulation, in the sense that the system dynamics are described by nonlinear continuous-time differential equations, and the nonlinear measurements are taken at intermittent discrete times randomly spaced, wherein the selection of a discretization time for the continuous-time differential model is unnecessary, which possesses wider applications, better accuracy, better stability, easier design, and easier implementation.
Another objective of the present invention is to provide a filtering process, wherein the propagation of the system state estimate between two consecutive measurement instants in the time update is governed by an adaptive stepsize control, which automatically determines the time step and guarantees the convergence.
Another objective of the present invention is to provide a filtering process, which validates measurement data, in order to reject poor-quality measurement data before they are fed into the filter.
Another objective of the present invention is to provide a filtering process, which corrects measurement data, in order to correct low-quality measurement data before they are fed into the filter.
Another objective of the present invention is to provide a filtering process, comprising an adaptive stepsize control to automatically compute the stepsize to propagate the backward state estimate in the backward time update.
Another objective of the present invention is to provide a filtering process, wherein square root implementations are enforced for covariance matrix propagation, wherever applied, to ensure numerical stability.
Another objective of the present invention is to provide a filtering process, wherein the system state estimates are obtained by using, selectively, forward estimating solution, and combined forward estimating solution and backward smoothing solution.
Accordingly, in order to accomplish the above objectives, the present invention provides a filtering process which comprises the steps of:
(a) computing a model relevant information for a time update and a measurement update in a model selection; and
(b) providing, from the model selection, an initial condition of a system estimate for a system model computation in the time update, an initial condition of a covariance error matrix for a derivative system model computation in the time update, a system model for the system model computation in the time update, a derivative system model for the derivative system model computation in the time update, a measurement model for a measurement model computation in the measurement update, and a derivative measurement model for a derivative measurement model computation in the measurement update.